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Sounds like your guy scared himself pretty good! I would definitely wait until all soreness from the vaccine is gone before saddling him again. I agree with Sabine on taking it slow and offering plenty of treats. I would also avoid saddling this goat anywhere but at home for now. If he suddenly decides to take off again there is less danger of him getting injured and you won't lose him. Home is a safer, more comfortable environment for any animal to learn new things.
Feel over his back, sides, and belly for any other possible sore spots besides the one you already know about. I agree with removing the britchen strap for now. Another thing to check is saddle fit. Is this goat particularly wide or narrow? Does he have good prominent withers, or do they sort of hide between the shoulder blades? His conformation can affect how your saddle fits, and if the tree is not right for his back then that could be what set him off in the first place. When you place the saddle on his back (unsecured), it should sit evenly front-to-back without rocking. Slide your hand from front to back underneath the saddle (palm down) and feel that the pressure is even down the length of the tree. Feel for places where you hand feels pinched or where there is space between the back of your hand and the saddle. Both are signs of a poorly fitting saddle or improper placement. See if your goat will walk with the saddle on but not cinched up. This could tell you more about the source of his fear/discomfort.
One other thing to check is that you are placing the saddle far back enough. It's easy to accidentally put the saddle too far forward where the front edge of the tree rubs the shoulder blades. This could frighten a skittish goat into bolting. A too-tight cinch can also scare a first-timer. Once you're sure that your goat is over his soreness and that the equipment fits and is adjusted correctly, I suggest saddling him up and leaving him loose with the herd until he gets used to it. Sometimes when it comes to desensitizing, a no-pressure approach is more effective than intentional training. You need to bond with this nervous goat if he's going to be a good trail companion, but for now that might better be saved for times when he's not wearing the saddle so he can fully concentrate on you.
Feel over his back, sides, and belly for any other possible sore spots besides the one you already know about. I agree with removing the britchen strap for now. Another thing to check is saddle fit. Is this goat particularly wide or narrow? Does he have good prominent withers, or do they sort of hide between the shoulder blades? His conformation can affect how your saddle fits, and if the tree is not right for his back then that could be what set him off in the first place. When you place the saddle on his back (unsecured), it should sit evenly front-to-back without rocking. Slide your hand from front to back underneath the saddle (palm down) and feel that the pressure is even down the length of the tree. Feel for places where you hand feels pinched or where there is space between the back of your hand and the saddle. Both are signs of a poorly fitting saddle or improper placement. See if your goat will walk with the saddle on but not cinched up. This could tell you more about the source of his fear/discomfort.
One other thing to check is that you are placing the saddle far back enough. It's easy to accidentally put the saddle too far forward where the front edge of the tree rubs the shoulder blades. This could frighten a skittish goat into bolting. A too-tight cinch can also scare a first-timer. Once you're sure that your goat is over his soreness and that the equipment fits and is adjusted correctly, I suggest saddling him up and leaving him loose with the herd until he gets used to it. Sometimes when it comes to desensitizing, a no-pressure approach is more effective than intentional training. You need to bond with this nervous goat if he's going to be a good trail companion, but for now that might better be saved for times when he's not wearing the saddle so he can fully concentrate on you.